The Sample Complexity of Revenue Maximizationeviterci/slides/SampleComplexityRM.pdf · The Sample...

The Sample Complexity

of Revenue Maximization in the Hierarchy of Deterministic Combinatorial Auctions

Ellen Vitercik

Joint work with Nina Balcan and Tuomas Sandholm

Theory Lunch

27 April 2016

Combinatorial (multi-item) auctions

Combinatorial auctions allow bidders to express preferences for bundles

of goods

: $5 : $5 : $6

Real-world examples

• US Government wireless spectrum auctions [FCC]

• Sourcing auctions [Sandholm 2013]

• Airport time slot allocation [Rassenti 1982]

• Building development, e.g. office space in GHC (no money)

• Property sales

• Mechanism designer must determine:

– Allocation function: Who gets what?

– Payment function: What does the auctioneer charge?

• Goal: design strategy-proof mechanisms

– Easy for the bidders to compute the optimal strategy

– Easy for designer to analyze possible outcomes

Mechanism design

Warm-up: single-item auctions

NINA

TUOMAS

: $5

: $3

Second-price auction: the classic strategy-proof, single-item auction.

Allocation (N:$5, T:$3)

= give carrot to Nina

Payment (N:$5, T:$3)

= charge Nina $3

, -$3

ø, -$0

Second-Price Auction

• Standard assumptions: bidders’ valuations drawn from

distribution 𝑫, mechanism designer knows 𝑫

– Allocation and payment rules often depend on 𝑫

Revenue-maximizing combinatorial auctions

• Central problem in Automated Mechanism Design

[Conitzer and Sandholm 2002, 2003, 2004, Likhodedov and

Sandholm 2004, 2005, 2015, Sandholm 2003]

Revenue-maximizing combinatorial auctions

No theory that relates the performance of the designed mechanism on the

samples to that mechanism’s expected performance on 𝑫, until now.

Design Challenges Feasible Solutions

Support of 𝐃 might be doubly-

exponential

Draw samples from 𝐃 instead

NP-hard to determine the

revenue-maximizing

deterministic auction with

respect to 𝐃

[Conitzer and Sandholm 2002]

Fix a rich class of auctions. Can

we learn the revenue-

maximizing combinatorial

auction in that class with

respect to 𝐃 given samples

drawn from 𝐃?

Outline

• Introduction

• Hierarchy of deterministic combinatorial auction

classes

• Our contribution: how many samples are needed to

learn over the hierarchy of auctions?

• Affine maximizer auctions and Rademacher complexity

• Mixed-bundling auctions and pseudo-dimension

• Summary and future directions

• 𝟑𝟐 possible outcomes 𝒐 = (𝒐𝟏, 𝒐𝟐)

• For example, 𝒐 = ( , )

Combinatorial auctions

: $1

: $0

: $1

: 50¢

: 50¢

: 50¢

NINA TUOMAS

• Social Welfare 𝒐

= SW 𝒐 = 𝒗𝒊 𝒐𝒊∈𝑩𝒊𝒅𝒅𝒆𝒓𝒔 𝒐∗ maximizes SW 𝒐

• SW-i 𝒐 = 𝒗𝒋 𝒐𝒋∈𝑩𝒊𝒅𝒅𝒆𝒓𝒔− 𝒊 𝒐−𝒊 maximizes SW-i 𝒐

• Allocation: 𝒐∗

• Payment: Nina pays SW −𝑵𝒊𝒏𝒂 𝒐−𝑵𝒊𝒏𝒂 − SW−𝑵𝒊𝒏𝒂 𝒐∗

A natural generalization of second price

The “Vickrey-Clarke-Groves mechanism” (VCG).

: $1

: $0

: $1

: 50¢

: 50¢

: 50¢

NINA TUOMAS

• 𝒐∗ = ,

• 𝒐−𝑵𝒊𝒏𝒂 = (∅, , )

• Nina pays 𝒗𝑻𝒖𝒐𝒎𝒂𝒔( , ) −𝒗𝑻𝒖𝒐𝒎𝒂𝒔( ) = 0

VCG in action

How do we get the bidders to pay more?

: $1

: $0

: $1

: 50¢

: 50¢

: 50¢

NINA TUOMAS

Outcome boosting

• value ∅, , = 𝒗𝑵𝒊𝒏𝒂 ∅ + 𝒗𝑻𝒖𝒐𝒎𝒂𝒔( , ) = 50¢

: $1

: $0

: $1

: 50¢

: 50¢

: 50¢

NINA TUOMAS

Outcome boosting

• value ∅, , = 𝒗𝑵𝒊𝒏𝒂 ∅ + 𝒗𝑻𝒖𝒐𝒎𝒂𝒔( , ) = 50¢ + 99¢

• 𝒐∗ = ,

• 𝒐−𝑵𝒊𝒏𝒂 = (∅, , )

: $1

: $0

: $1

: 50¢

: 50¢

: 50¢

NINA TUOMAS

Outcome boosting


• 𝒐∗ = ,

• 𝒐−𝑵𝒊𝒏𝒂 = (∅, , )

• Nina pays 𝒗𝑻𝒖𝒐𝒎𝒂𝒔( , ) + 99¢ −𝒗𝑻𝒖𝒐𝒎𝒂𝒔( ) = 99¢

: $1

: $0

: $1

: 50¢

: 50¢

: 50¢

NINA TUOMAS

• Boost outcomes: 𝝀(𝒐)

• Take bids 𝒗

• Compute outcome:

𝒏

𝒋∈𝑩𝒊𝒅𝒅𝒆𝒓𝒔

𝝀 𝒐∗ = 𝒂𝒓𝒈𝒎𝒂𝒙𝒐 𝑺𝑾 𝒐 + 𝝀 𝒐

• Compute Bidder 𝒊’s payment:

𝑺𝑾−𝒊 𝒐−𝒊 + 𝝀 𝒐−𝒊 − 𝑺𝑾−𝒊 𝒐

∗ + 𝝀 𝒐∗

Affine maximizer auctions (AMAs)

• Boost outcomes: 𝝀(𝒐)

• Take bids 𝒗


𝒐∗ = 𝒂𝒓𝒈𝒎𝒂𝒙 𝒐

𝒗𝒋 𝒐 +

𝒏


𝝀 𝒐


𝒗𝒋 𝒐−𝒊

𝒋∈𝑩𝒊𝒅𝒅𝒆𝒓𝒔−{𝒊}

+ 𝝀 𝒐−𝒊 − 𝒗𝒋 𝒐∗


+ 𝝀 𝒐∗


• Boost outcomes: 𝝀(𝒐); Weight bidders: 𝒘𝒊

• Take bids 𝒗



𝒗𝒋 𝒐 +

𝒏


𝝀 𝒐






+ 𝝀 𝒐∗



• Take bids 𝒗



𝒘𝒋𝒗𝒋 𝒐 +

𝒏


𝝀 𝒐






+ 𝝀 𝒐∗



• Take bids 𝒗




𝒏


𝝀 𝒐


𝟏

𝒘𝒊 𝒘𝒋𝒗𝒋 𝒐

−𝒊


+ 𝝀 𝒐−𝒊 − 𝒘𝒋𝒗𝒋 𝒐∗


+ 𝝀 𝒐∗


Hierarchy of parameterized auction classes

Affine maximizer auctions [R79] 𝒘𝒊, 𝝀 𝒐 ∈ ℝ

Virtual valuation

combinatorial auctions

[SL03]

𝝀 𝒐 = 𝝀𝒊 𝒐

𝒊∈𝑩𝒊𝒅𝒅𝒆𝒓𝒔

𝝀-auctions [J07] • 𝒘𝒊 = 𝟏 • 𝝀 𝒐 ∈ ℝ

Mixed bundling auctions

with reserve prices [TS12]

• 𝒘𝒊 = 𝟏 • 𝝀 𝒐 = 𝟎 except any

outcome where a

bidder gets all items

• item reserve prices

∪ ∪

∪ ∪

∪ • 𝒘𝒊 = 𝟏 • 𝝀 𝒐 = 𝟎 except

outcome where a

bidder gets all items

Mixed bundling auctions

[J07]

Outline

• Introduction


classes






• Optimize 𝝀 𝒐 and 𝒘 given a sample 𝑺~𝑫𝑵

– (Automated Mechanism Design)

• We want:

– The auction with best revenue over the sample has almost optimal expected revenue

– Any approximately revenue-maximizing auction over the sample will have approximately optimal expected revenue

• For any auction we output, we want |𝑺| large enough such that:

|empirical revenue – expected revenue| < 𝝐

• In other words, how many samples 𝐒 = 𝑵 do we need to ensure that

|empirical revenue – expected revenue|

=𝟏

𝑵 𝒓𝒆𝒗𝑨 𝒗

𝒗∈𝑺

− 𝔼𝒗~𝑫 𝒓𝒆𝒗𝑨 𝒗 < 𝝐

for all auctions 𝑨 in the class?

• (We can only do this with high probability.)

Our contribution

How many samples do we need?

Affine maximizer auctions [R79]

𝑵 = 𝑶 𝑼𝒏𝒎 𝒎 𝑼+ 𝒏𝒎/𝟐 /𝝐𝟐

Virtual valuation combinatorial auctions

[SL03]


𝝀-auctions [J07]


Mixed bundling auctions [J07]

𝑵 = 𝑶 𝑼/𝝐 𝟐

Mixed bundling auctions with reserve prices

[TS12]

𝑵 = 𝑶 𝑼/𝝐 𝟐𝒎𝟑

∪ ∪

∪ ∪

∪

Variables

𝑵: sample size

𝒏: # bidders

𝒎: # items

𝑼: maximum revenue

achievable over the

support of the

bidders’ valuation

distributions





[SL03]


𝝀-auctions [J07]



𝑵 = 𝑶 𝑼/𝝐 𝟐


[TS12]

𝑵 = 𝑶 𝑼/𝝐 𝟐𝒎𝟑 Variables

𝑵: sample size

𝒏: # bidders

𝒎: # items


achievable over the

support of the


distributions

∪ ∪

∪ ∪

∪

Nearly-matching exponential lower bounds.





[SL03]


𝝀-auctions [J07]



𝑵 = 𝑶 𝑼/𝝐 𝟐


[TS12]

𝑵 = 𝑶 𝑼/𝝐 𝟐𝒎𝟑

∪ ∪

∪ ∪

∪

Variables

𝑵: sample size

𝒏: # bidders

𝒎: # items


achievable over the

support of the


distributions

Learning theory tool: Rademacher complexity





[SL03]


𝝀-auctions [J07]



𝑵 = 𝑶 𝑼/𝝐 𝟐


[TS12]

𝑵 = 𝑶 𝑼/𝝐 𝟐𝒎𝟑

∪ ∪

∪ ∪

∪

Variables

𝑵: sample size

𝒏: # bidders

𝒎: # items


achievable over the

support of the


distributions

Learning theory tool: Pseudo-dimension

Outline

• Introduction


classes










[SL03]


𝝀-auctions [J07]



𝑵 = 𝑶 𝑼/𝝐 𝟐


[TS12]

𝑵 = 𝑶 𝑼/𝝐 𝟐𝒎𝟑

∪ ∪

∪ ∪

∪

Variables

𝑵: sample size

𝒏: # bidders

𝒎: # items


achievable over the

support of the


distributions


• Take bids 𝒗




𝒏


𝝀 𝒐


𝟏

𝒘𝒊 𝒘𝒋𝒗𝒋 𝒐

−𝒊


+ 𝝀 𝒐−𝒊 − 𝒘𝒋𝒗𝒋 𝒐∗


+ 𝝀 𝒐∗

Key challenge

Our

problem...

Whereas

typically in

machine

learning…

• More expressive function classes need more samples to learn

• How to measure expressivity?

– How well do functions from the class fit random noise?

• Empirical Rademacher complexity:

𝒙𝟏, … , 𝒙𝑵 ~ −𝟏, 𝟏 𝑵, 𝑺 = 𝒗𝟏, … , 𝒗𝑵

𝑹𝑺 𝓐 = 𝔼𝒙 𝒔𝒖𝒑𝑨∈𝓐𝟏

𝑵 𝒙𝒊 ∙ 𝒓𝒆𝒗𝑨 𝒗𝒊 , where

• Rademacher complexity:

𝑹𝑵 𝓐 = 𝔼𝑺~𝑫𝑵 𝑹𝑺 𝓐

Rademacher complexity

• With probability at least 𝟏 − 𝜹, for all 𝑨 ∈ 𝓐,

|empirical revenue – expected revenue| ≤ 𝟐𝑹𝑵 𝓐 +𝑼𝟐 𝒍𝒏 𝟐/𝜹

𝑵

*𝑼 is the maximum revenue achievable over the support of the bidders’ valuation distributions

• More expressive function classes need more samples to learn

• How to measure expressivity?

– How well do functions from the class fit random noise?

• Empirical Rademacher complexity:

𝒙𝟏, … , 𝒙𝑵 ~ −𝟏, 𝟏 𝑵, 𝑺 = 𝒗𝟏, … , 𝒗𝑵

𝑹𝑺 𝓐 = 𝔼𝒙 𝒔𝒖𝒑𝑨∈𝓐𝟏

𝑵 𝒙𝒊 ∙ 𝒓𝒆𝒗𝑨 𝒗𝒊 , where

• Rademacher complexity:

𝑹𝑵 𝓐 = 𝔼𝑺~𝑫𝑵 𝑹𝑺 𝓐

Rademacher complexity

𝓐 = all binary valued functions 𝑹𝑵 𝓐 =

𝟏

𝟐

𝓐 = one binary valued function 𝑹𝑵 𝓐 = 𝟎

• Key idea: split revenue function into its simpler components

– Weighted social welfare without any one bidder’s

participation (𝒏 components)

– Amount of revenue subtracted out to maintain strategy-

proof property

• Then use compositional properties of Rademacher complexity

and other tricks, for example:

If 𝑭 = 𝒇 | 𝒇 = 𝒈 + 𝒉,𝒈 ∈ 𝑮, 𝒉 ∈ 𝑯 , then 𝑹𝑵 𝑭 ≤ 𝑹𝑵 𝑮 + 𝑹𝑵 𝑯

Rademacher complexity of AMAs

Let 𝓐 be the class of 𝒏-bidder, 𝒎-item AMA revenue functions. If


,

then with high probability over a sample 𝑺~𝑫𝑵,

|empirical revenue – expected revenue| < 𝝐 for all 𝒓𝒆𝒗𝑨 ∈ 𝓐.

Theorem

Outline

• Introduction


classes










[SL03]


𝝀-auctions [J07]



𝑵 = 𝑶 𝑼/𝝐 𝟐


[TS12]

𝑵 = 𝑶 𝑼/𝝐 𝟐𝒎𝟑

Variables

𝑵: sample size

𝒏: # bidders

𝒎: # items


achievable over the

support of the


distributions

∪ ∪

∪ ∪

∪

• Class of auctions parameterized by a scalar 𝒄

• Boost the allocations where one bidder gets all goods by 𝒄


• value , , ∅ = 𝒗𝑵𝒊𝒏𝒂 , + 𝒗𝑻𝒖𝒐𝒎𝒂𝒔(∅) = 50¢ + 99¢

• How large must the sample 𝑺 be in order to ensure that for all MBAs, |empirical revenue – expected revenue| < 𝝐?

Mixed bundling auctions (MBAs)

: $1

: $0

: $1

: 50¢

: 50¢

: 50¢

NINA TUOMAS

𝒄

Structural properties of MBA revenue functions

Fix 𝒗 ∈ 𝑺. Then 𝒓𝒆𝒗𝒗 𝒄 is piecewise linear with at most 𝒏 + 𝟏

discontinuities.

Lemma re

ve

nu

e

• Complexity measure for binary-valued functions only

• Example: 𝑭 = {single interval on the real line}

• No set of size 3 can be labeled in all 𝟐𝟑 ways by 𝑭

VC-dimension

How can we extend VC-dim to real-valued functions?

+ + -

• Class of functions 𝑭 shatters set 𝑺 = 𝒙𝟏, … , 𝒙𝑵 if

for all 𝐛 ∈ 𝟎, 𝟏 𝑵, there exists 𝒇 ∈ 𝑭 such that 𝒇 𝒙𝒊 = 𝒃𝒊 • VC-dimension of 𝑭 is the cardinality of the largest set 𝑺 that can be shattered by 𝑭

• Sample 𝑺 = 𝒙𝟏, … , 𝒙𝑵

• Class of functions 𝑭 into −𝑼,𝑼

• 𝒓 = 𝒓(𝟏), … , 𝒓(𝑵) ∈ ℝ𝑵

witnesses the shattering of 𝑺 by 𝑭

if for all 𝑻 ⊆ 𝑺, there exists 𝒇𝑻 ∈ 𝑭

such that 𝒇𝑻 𝒙𝒊 ≤ 𝒓(𝒊) iff 𝒙𝒊 ∈ 𝑻

• Pseudo-dimension of 𝑭 is the

cardinality of the largest sample 𝑺 that can be shattered by 𝑭

Pseudo-dimension

P-dim(𝑭) = VC-dim( (𝒙, 𝒓) ⟼ 𝟏𝒇 𝒙 −𝒓>𝟎| 𝒇 ∈ 𝑭 )

𝒙𝟏 𝒇 𝒙𝟏 ≤ 𝒓(𝟏) 𝟎

𝒙𝟐 𝒇 𝒙𝟐 ≤ 𝒓(𝟐) 𝟎

𝒙𝟑 𝒇 𝒙𝟑 > 𝒓(𝟑) 𝟏

𝒙𝟒 𝒇 𝒙𝟒 ≤ 𝒓(𝟒) 𝟎

𝒙𝟓 𝒇 𝒙𝟓 > 𝒓(𝟓) 𝟏

𝒙𝟔 𝒇 𝒙𝟔 ≤ 𝒓(𝟔) 𝟎

𝒙𝟕 𝒇 𝒙𝟕 > 𝒓(𝟕) 𝟏

𝒙𝟖 𝒇 𝒙𝟖 > 𝒓(𝟖) 𝟏

𝒙𝟗 𝒇 𝒙𝟗 ≤ 𝒓(𝟗) 𝟎

• Set of auction revenue functions 𝓐 with range in 𝟎, 𝑼 ,

distribution 𝑫 over valuations 𝒗.

• For every 𝝐 > 𝟎, 𝜹 ∈ 𝟎, 𝟏 , if

𝑵 = 𝑶𝑼

𝝐

𝟐P−dim(𝓐) ∗ 𝐥𝐧

𝑼

𝝐+ 𝐥𝐧

𝟏

𝜹,

then with probability at least 𝟏 − 𝜹 over a sample 𝑺~𝑫𝑵,

|empirical revenue – expected revenue| < 𝝐

for every 𝒓𝒆𝒗𝑨 ∈ 𝓐.


Pseudo-dimension allows us to derive strong sample complexity bounds.





[SL03]


𝝀-auctions [J07]



𝑵 = 𝑶 𝑼/𝝐 𝟐


[TS12]

𝑵 = 𝑶 𝑼/𝝐 𝟐𝒎𝟑

Variables

𝑵: sample size

𝒏: # bidders

𝒎: # items


achievable over the

support of the


distributions

∪ ∪

∪ ∪

∪

2-bidder MBA pseudo-dimension

Let 𝓐 = 𝒓𝒆𝒗𝒄 𝒄≥𝟎 be the class of 𝒏-bidder, 𝒎-item mixed bundling

auction revenue functions. Then P-dim(𝓐) = 𝑶 𝐥𝐨𝐠𝒏 .

Theorem

Proof sketch.

• Fact: there exists a set of 2 samples that is shattered by 𝓐.

• Suppose, for a contradiction, that 𝑺 = 𝒗𝟏, 𝒗𝟐, 𝒗𝟑 is shatterable.

• Recall 𝒗𝟏 = 𝒗𝟏𝟏, 𝒗𝟐

𝟏

• This means:

– There exists 𝒓 = 𝒓𝟏, 𝒓𝟐, 𝒓𝟑 ∈ ℝ𝟑 and

𝟐|𝑺| = 𝟖 MBA parameters 𝐂 = 𝒄𝟏, … , 𝒄𝟖 such that

𝒓𝒆𝒗𝒄𝟏 , … , 𝒓𝒆𝒗𝒄𝟖 induce all 8 binary labelings on 𝑺 with respect to 𝒓.


Let 𝓐 = 𝒓𝒆𝒗𝒄 𝒄≥𝟎 be the class of 𝟐-bidder, 𝒎-item mixed bundling

auction revenue functions. Then P-dim(𝓐) = 𝟐.

Theorem

We need to show that no set of 3 samples can be shattered by 𝓐.

𝐦𝐢𝐧 𝒗𝟏𝒊 𝒎 ,𝒗𝟐

𝒊 𝒎

𝒄𝒊 𝒄


Fix 𝒗𝒊 ∈ 𝑺. Then 𝒓𝒆𝒗𝒗𝒊 𝒄 is piecewise linear with one discontinuity,

with a slope of 2 followed by a constant function with value

𝐦𝐢𝐧 𝒗𝟏𝒊 𝒎 ,𝒗𝟐

𝒊 𝒎 .

Lemma

reve

nu

e

𝐦𝐢𝐧 𝒗𝟐𝟑 𝒎 ,𝒗𝟐

𝟑 𝒎

𝒄𝒊 𝒄

Case 1: 𝒓𝟑 < 𝒎𝒊𝒏 𝒗𝟏𝟑 𝒎 ,𝒗𝟐

𝟑 𝒎

𝒓𝟑 reve

nu

e


𝟑 𝒎

𝒓𝒆𝒗𝒗𝟑(𝒄) increasing 𝒓𝒆𝒗𝒗𝟑 𝒄 = 𝐦𝐢𝐧 𝒗𝟏𝟑 𝒎 ,𝒗𝟐

𝟑 𝒎

𝒄𝟑

𝒄


𝟑 𝒎


𝟑 𝒎

𝒓𝒆𝒗𝒗𝟐(𝒄) increasing 𝒓𝒆𝒗𝒗𝟐 𝒄 = 𝐦𝐢𝐧 𝒗𝟏𝟐 𝒎 ,𝒗𝟐

𝟐 𝒎

𝒄𝟑 𝒄𝟐

𝒄


𝟑 𝒎


𝟑 𝒎


𝟐 𝒎

𝒓𝒆𝒗𝒗𝟏(𝒄) increasing 𝒓𝒆𝒗𝒗𝟏 𝒄 = 𝐦𝐢𝐧 𝒗𝟏𝟏 𝒎 ,𝒗𝟐

𝟏 𝒎

𝒄𝟑 𝒄𝟐 𝒄𝟏

𝒄


𝟑 𝒎


𝟐 𝒎

𝒓𝒆𝒗𝒗𝟏(𝒄) increasing 𝒓𝒆𝒗𝒗𝟏 𝒄 = 𝐦𝐢𝐧 𝒗𝟏𝟏 𝒎 ,𝒗𝟐

𝟏 𝒎

𝒄𝟑 𝒄𝟐 𝒄𝟏

𝒄


𝟑 𝒎

𝒓𝒆𝒗𝒗𝟐(𝒄) increasing

𝒓𝒆𝒗𝒗𝟏(𝒄) increasing

We need:

This is impossible, so we reach

a contradiction. Therefore, no

set of size 3 can be shattered

by the class of 2-bidder MBA

revenue functions, so the

pseudo-dimension is at most 2.

𝒓𝒆𝒗𝒗𝟑(𝒄) increasing

Summary

• Analyzed the sample complexity of learning over a

hierarchy of deterministic combinatorial auctions

• Uncovered structural properties of these auctions’

revenue functions along the way

– Of independent interest beyond sample complexity

results

The Sample Complexity of Revenue Maximizationeviterci/slides/SampleComplexityRM.pdf · The Sample...

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